On Growth Functions of Coxeter Groups

2024/05/17 by Bischof, Sebastian
#20F55 #51F15 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2405.10617

Abstract

Let (W, S) be a Coxeter system of rank n and let p_{(W, S)}(t) be its growth function. It is known that p_{(W, S)}(q^-1) < ∞ holds for all n ≤ q ∈ ℕ. In this paper we will show that this still holds for q = n-1, if (W, S) is 2-spherical. Moreover, we will prove that p_{(W, S)}(q^-1) = ∞ holds for q = n-2, if the Coxeter diagram of (W, S) is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of 2-spherical Coxeter systems with complete Coxeter diagram.

Related

• On the continuity of the growth rate on the space of Coxeter systems
• On growth types of quotients of Coxeter groups by parabolic subgroups
• Growth rates of Coxeter groups and Perron numbers
• On the growth of cocompact hyperbolic Coxeter groups
• Data about hyperbolic Coxeter systems